Reflections of Graphs

students, graphs can reveal a lot about how quantities change, and one of the most useful transformations is a reflection ✨. A reflection flips a graph over a line, like seeing it in a mirror. In IB Mathematics Analysis and Approaches SL, reflections help you understand how a function changes when its input or output is reversed. This lesson will show you the main ideas, the notation, and how reflections connect to the broader topic of functions.

What is a reflection?

A reflection is a transformation that creates a mirror image of a graph across a line. In coordinate geometry, the most common reflection lines are the $x$-axis, the $y$-axis, and the line $y=x$. When a graph is reflected, the shape stays the same, but its position changes. This is important because the graph still represents a relationship, but the relationship has been transformed.

For a function $y=f(x)$:

Reflection in the $x$-axis gives $y=-f(x)$.
Reflection in the $y$-axis gives $y=f(-x)$.
Reflection in the line $y=x$ swaps the roles of $x$ and $y$, so the reflected graph is related to the inverse function when one exists.

A key idea is that the reflection changes either the output values, the input values, or both. This is why reflections are studied alongside translations, stretches, and compressions in the Functions topic.

Reflection in the $x$-axis

Reflecting a graph in the $x$-axis changes every point $(x,y)$ to $(x,-y)$. For a function, this means each output is multiplied by $-1$, so the new function is $y=-f(x)$.

This reflection makes positive values negative and negative values positive. If the original graph is above the $x$-axis, the reflected graph appears below it, and vice versa.

Example: If $f(x)=x^2$, then the reflection in the $x$-axis is $y=-x^2$. The original graph is a parabola opening upward, while the reflected graph opens downward. Both graphs have the same vertex at $(0,0)$, but their direction is reversed.

Another example: If $f(x)=2x+1$, then the reflected graph is $y=-(2x+1)=-2x-1$. Here, the line still has slope magnitude $2$, but the graph is flipped vertically.

This kind of reflection is useful when modeling situations where one quantity is the negative of another, such as a debt instead of a balance, or downward motion compared with upward motion 📉.

Reflection in the $y$-axis

Reflecting a graph in the $y$-axis changes each point $(x,y)$ to $(-x,y)$. For functions, this means replacing $x$ with $-x$, giving $y=f(-x)$.

This reflection is often called a horizontal reflection because the graph is flipped left-to-right. It does not change output values, but it changes which input produces each output.

Example: If $f(x)=x^3$, then $f(-x)=(-x)^3=-x^3$. In this case, the reflected graph is the same as multiplying by $-1$, because the function is odd. That is a special property of some functions, not a rule for all functions.

Example: If $f(x)=x^2+2x$, then the reflected graph is $f(-x)=(-x)^2+2(-x)=x^2-2x$. This is a different parabola with the same shape, but it is mirrored across the $y$-axis.

A common mistake is to confuse $f(-x)$ with $-f(x)$. These are not the same in general. The first reflects in the $y$-axis, while the second reflects in the $x$-axis. For example, if $f(x)=x^2+1$, then $f(-x)=(-x)^2+1=x^2+1$, but $-f(x)=-(x^2+1)=-x^2-1$.

Reflection in the line $y=x$ and inverse functions

The reflection of a graph in the line $y=x$ is especially important in the study of inverse functions. Every point $(x,y)$ becomes $(y,x)$. This means the $x$- and $y$-coordinates switch places.

If a function has an inverse, then the graph of the inverse function is the reflection of the original graph in the line $y=x$. This happens only when the function is one-to-one, meaning each output comes from exactly one input.

Example: Let $f(x)=x^3$. Its inverse is f^{-1}(x)=

oot 3

\of{x}$. The graph of $y=$\root 3$ \of{x}$ is the mirror image of $y=x^3$ across the line $y=x.

Example: Let $f(x)=e^x$. Its inverse is $f^{-1}(x)=\ln x$. These two graphs are reflections of each other across $y=x$.

This connection is very useful in IB Mathematics because inverse functions are built from the idea of reversing a relationship. If the graph of a function is reflected in $y=x$, the new graph shows what happens when the input and output are swapped 🔁.

How to describe and sketch reflections

When sketching reflections, always identify the line of reflection first. Then use point mapping rules.

For reflection in the $x$-axis:

$(x,y) \to (x,-y)$
algebraic rule: $y=-f(x)$

For reflection in the $y$-axis:

$(x,y) \to (-x,y)$
algebraic rule: $y=f(-x)$

For reflection in the line $y=x$:

$(x,y) \to (y,x)$
algebraic idea: interchange $x$ and $y$

Suppose a point on a graph is $(2,3)$. Its reflection in the $x$-axis is $(2,-3)$, in the $y$-axis is $(-2,3)$, and in the line $y=x$ is $(3,2)$.

When sketching from a function rule, it is helpful to think about key points such as intercepts, turning points, and asymptotes. Reflections move these features in predictable ways. For example, the $x$-intercepts stay on the $x$-axis after reflection in the $x$-axis because $y=0$ remains $y=0$. But a $y$-intercept may change sign in an $x$-axis reflection.

Reflections with different types of functions

Reflections appear across many function types in IB Mathematics.

For a linear function $f(x)=mx+c$:

$-f(x)=-mx-c$ reflects across the $x$-axis.
$f(-x)=-mx+c$ reflects across the $y$-axis.

For a quadratic function $f(x)=ax^2+bx+c$:

$-f(x)=-ax^2-bx-c$ flips the parabola vertically.
$f(-x)=ax^2-bx+c$ reflects it across the $y$-axis.

For a rational function such as $f(x)=\frac{1}{x}$:

$-f(x)=-\frac{1}{x}$ reflects in the $x$-axis.
$f(-x)=\frac{1}{-x}=-\frac{1}{x}$ reflects in the $y$-axis, and here the result is the same as $-f(x)$ because the function is odd.

For an exponential function $f(x)=2^x$:

$f(-x)=2^{-x}=\frac{1}{2^x}$ reflects in the $y$-axis.
$-f(x)=-2^x$ reflects in the $x$-axis.

For a logarithmic function $f(x)=\ln x$:

$-f(x)=-\ln x$ reflects in the $x$-axis.
$f(-x)=\ln(-x)$ is not defined for real $x>0$ in the same domain, so reflections can affect domain limits and graph visibility.

These examples show that reflections are not just visual tricks. They affect algebra, domain, range, and function behavior.

Common errors to avoid

One common error is thinking that $f(-x)$ and $-f(x)$ mean the same thing. They do not. The minus sign inside the function changes the input, while the minus sign outside changes the output.

Another error is forgetting that reflections can change the domain. For example, if $f(x)=\ln x$, then the reflected graph in the $y$-axis would involve $\ln(-x)$, which only makes sense when $x<0$. So the reflected graph appears on the left side of the coordinate plane.

A third error is assuming that a reflected graph must look completely different. Often the graph has the same shape, just flipped. This is especially true for symmetric functions. For instance, if $f(x)=x^2$, then $f(-x)=x^2$, so the graph is unchanged because it is symmetric about the $y$-axis.

Conclusion

Reflections of graphs are a core part of transformations in the Functions topic. students, you should remember that reflecting in the $x$-axis gives $y=-f(x)$, reflecting in the $y$-axis gives $y=f(-x)$, and reflecting in the line $y=x$ swaps coordinates and connects directly to inverse functions. These ideas help you read, sketch, and compare graphs more accurately. In IB Mathematics Analysis and Approaches SL, reflections are important because they strengthen your understanding of how functions are represented, transformed, and related to one another.

Study Notes

A reflection creates a mirror image of a graph across a line.
Reflection in the $x$-axis maps $(x,y)$ to $(x,-y)$ and gives $y=-f(x)$.
Reflection in the $y$-axis maps $(x,y)$ to $(-x,y)$ and gives $y=f(-x)$.
Reflection in the line $y=x$ maps $(x,y)$ to $(y,x)$ and is linked to inverse functions.
$f(-x)$ and $-f(x)$ are different in general.
Reflections affect domain and range, especially for functions like $\ln x$ and rational functions.
Symmetric functions may look unchanged after a reflection.
Reflections are part of the broader study of transformations in functions, alongside shifts and stretches.